Definition:Congruence (Geometry)/Warning

Congruence: Warning

Two geometric figures which are congruent are not necessarily identical.

For example, consider two scalene triangles with identical sides and identical internal angles embedded in the plane.

These scalene triangles are not identical if one is a reflection of the other.

However, they are still congruent, because they can be rotated in space $180 \degrees$ about an axis in the plane in which they are both embedded.

This has the same effect of picking one triangle up, flipping it over, and placing it down again the other way round.

In three-dimensional space, this point is important because mirror images cannot be superimposed by physically manipulating them in space.

Direct Congruence

Let $A$ and $B$ be $3$-dimensional figures which are congruent.

$A$ and $B$ are directly congruent if and only if $A$ and $B$ can be made to coincide with rotations and translations.

Opposite Congruence

Let $A$ and $B$ be $3$-dimensional figures which are congruent.

$A$ and $B$ are oppositely congruent if and only if $A$ and $B$ cannot be made to coincide with rotations and translations, but also need a reflection for this to happen.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): congruent: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): congruent: 1.